In this paper, a new bounded distribution is introduced and some distributional properties of the new distribution are discussed. Moreover, the new distribution is implemented in the field of engineering to the Cpc process capability index. Three unknown parameters of the distribution are estimated with several estimators, and the performances of the estimators are evaluated with a Monte Carlo simulation. A new regression model is introduced based on this new distribution as an alternative to beta and Kumaraswamy models. Furthermore, it is considered one of the first studies where regression model parameters are estimated using least squares, weighted least squares, Cramér–von Mises, and maximum product spacing estimators other than the maximum likelihood. The efficiency of the estimators for the parameters of the regression model is further assessed through a simulation. Real datasets are analyzed to demonstrate the applicability of the new distribution and regression model.